OFFSET
0,2
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for linear recurrences with constant coefficients, signature (3,3,-9).
FORMULA
G.f.: (1+3*x)*(1-x) / ((1-3*x)*(1-3*x^2)).
From Colin Barker, Feb 04 2017: (Start)
a(n) = 2*3^n - 3^((n-1)/2)/2*(1-(-1)^n+sqrt(3) + (-1)^n*sqrt(3)).
a(n) = 3*a(n-1) + 3*a(n-2) - 9*a(n-3) for n>2.
(End)
MATHEMATICA
A255442[n_] := SeriesCoefficient[(1 + 3*x)*(1 - x)/((1 - 3*x)*(1 - 3*x^2)), {x, 0, n}]; Array[A255442, 30, 0] (* JungHwan Min, Sep 29 2016 *)
A255442L[n_] := CoefficientList[Series[(1 + 3*x)*(1 - x)/((1 - 3*x)*(1 - 3*x^2)), {x, 0, n}], x]; A255442L[29] (* JungHwan Min, Sep 29 2016 *)
PROG
(PARI) Vec((1+3*x)*(1-x) / ((1-3*x)*(1-3*x^2)) + O(x^30)) \\ Colin Barker, Feb 04 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane and Doron Zeilberger, Feb 23 2015
STATUS
approved